Yes, that does indeed sound like something I might have said :)

I was referring to some extremely powerful theorems, originally due to Michael Harris, which show that:
- The cohomology groups of automorphic vector bundles on toroidal compactifications of Shimura varieties are isomorphic to certain spaces of (usually non-holomorphic) automorphic forms. (These vector bundles are coherent sheaves, hence "coherent cohomology".)
- Via this isomorphism, the Serre duality pairing on coherent cohomology groups is given by integration of automorphic forms.

The general results are all set out in Harris' paper "[Automorphic forms of $\overline{\partial}$-cohomology type as coherent cohomology classes][1]" (J. Diff. Geom. 1990); the relation between Serre duality and integrals is Proposition 3.8. 

To get some feel for how this is applied in practice I recommend reading some other papers of Harris from around the same time, e.g. this one with Kudla: [Arithmetic automorphic forms for the nonholomorphic discrete series of GSp(2)][2].

  [1]: https://doi.org/10.4310/jdg/1214445036
  [2]: https://doi.org/10.1215/S0012-7094-92-06603-8